Constrained Training Manifolds

Manifold: A continuous surface embedded within high-dimensional space where local regions resemble Euclidean space.
Constraint mapping: Instead of updating weights arbitrarily, optimization steps are projected onto the manifold, ensuring updates respect structural assumptions (e.g., low-rank approximations, orthogonality, symmetry).
Benefits: Improved training stability, reduced mode collapse, better transfer across tasks, and explicit safety guarantees when manifolds encode prohibited behaviors.

Manifold Type	Intuition	Use Cases
Low-rank subspaces	Restrict weight matrices to low-rank decompositions	Memory savings, faster adaptation, reduced overfitting
Orthogonal transforms	Preserve energy and minimize distortion	Stable recurrent or attention blocks, improved gradient flow
Sparsity-constrained surfaces	Maintain structured zeros	Interpretability, runtime efficiency
Policy-safe regions	Encode forbidden behavior vectors	Safety-aligned fine-tuning

Conceptual primer: What is a training manifold?